A SUITE OF SHARP INTERFACE MODELS FOR SOLID …ta.twi.tudelft.nl/nw/users/vuik/papers/Ver06PVZ.pdf · A SUITE OF SHARP INTERFACE MODELS FOR ... Abstract. A suite of mathematical models

European Conference on Computational Fluid DynamicsECCOMAS CFD 2006

P. Wesseling, E. Onate and J. Periaux (Eds)c© TU Delft, The Netherlands, 2006

A SUITE OF SHARP INTERFACE MODELS FORSOLID-STATE TRANSFORMATIONS IN ALLOYS

F.J. Vermolen, P. Pinson, C. Vuik∗, S. van der Zwaag†

∗Delft University of Technology, Delft Institute of Applied Mathematics,Mekelweg 4, 2628 CD, The Netherlands

e-mail: [email protected] page: http://ta.twi.tudelft.nl/users/vermolen

†Delft University of Technology, Department of Aerospace Engineering,Kluyverweg 1, 2629 HS Delft, The Netherlands

Key words: Moving boundary problem, Phase transformation

Abstract. A suite of mathematical models for solid-state phase transformations in

solid-state alloys is presented. The work deals in particular with dissolution and growth

of particles in solid-state alloys. Several models are presented and numerical solution pro-

cedures are summarized briefly. Further, some analytic solutions are considered shortly,

which are useful to validate the numerical solutions. One model, which incorporates sur-

face energy effects is presented in more detail. From this model it is observed that the

surface energy prevents the occurrence of fingering during growth.

1 INTRODUCTION

1.1 Technological background

In the thermal processing of both ferrous and non-ferrous alloys, homogenization of theexisting microstructure by annealing at such a high temperature that unwanted precipi-tates are fully dissolved, is required to obtain a microstructure suited to undergo heavyplastic deformation as well as providing an optimal starting condition for a subsequent pre-cipitation hardening treatment. Such a homogenization treatment is for example appliedprior to hot-rolling of Al killed construction steels, HSLA steels, all engineering steels, aswell as in processing aluminium extrusion alloys. Although precipitate dissolution is notthe only metallurgical process taking place during homogenization, it is often the mostcritical of the processes occurring. The minimum temperature at which the annealingshould take place can be determined from thermodynamic analysis of the phases present.However, the minimum annealing time at this temperature is not a constant but dependson particle size, particle geometry, particle concentration, overall composition etc. Tomake the homogenization treatment more efficient, it is highly desirable to have robustphysical models for the kinetics of particle dissolution as a function of thermodynamics

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F.J. Vermolen, P. Pinson, C. Vuik and S. van der Zwaag

and thermokinetics data as well as particle morphology and microstructural dimensions.Using such models the minimum annealing times and optimum heating strategics can becalculated a priori, rather than be determined experimentally, and at great cost.

Apart from their technological relevance, accurate physical models for particle dissolu-tion are, due to the complexity of the processes, also of great scientific and mathematicalinterest in themselves.

1.2 Existing models for particle dissolution

To describe particle dissolution several older mathematical models have been developed,which incorporate long-range diffusion [1, 2, 3] and non-equilibrium conditions at theinterface [4, 5]. In general, the dissolution of particles proceeds via decomposition of thechemical compound, the crossing of the atoms of the interface and long-range diffusionin the matrix. The first two processes are referred to as the interfacial processes. Thelong-range diffusion models are based on the assumption that the interfacial processesare infinitely fast. Hence, these models provide an upper boundary for the dissolutionkinetics.

The first models were based on analytical solutions for the interfacial position as afunction of time (see for instance Whelan [1] and Crank [6]). However, in these solutionsthe volume in which dissolution takes place is infinite. As far as we know, Baty, Tanzilliand Heckel were the first authors in the metallurgical community to applied a FiniteDifference Model [2] where the volume is bounded. Tundal and Ryum [3] also applied aFinite Difference Model in which a lognormal particle size distribution is included. Theyshowed that the macroscopic dissolution rates depend strongly on the particle size andpossible interactions between subsequent particles. Nolfi’s [4] model was, as far as known,the first model in which non-equilibrium conditions at the particle-matrix interface wereincluded. However, the interface migration was not included. The non-equilibrium con-dition is modelled by a Robin-condition at the interface. Their solution is in terms of aFourier series. Aaron and Kotler [5] combine Whelan’s solution with the incorporation ofthe Gibbs-Thomson effect to deal with the influence of curvature on the interface motion.Further, they transform the Robin-boundary condition of Nolfi’s model into a Dirichletboundary condition. Recently, Svoboda et al. [7] analyzed the kinetics of diffusionaltransformations where mechanical and chemical forces exerted on interfaces between sub-sequent phases are incorporated. Their approach is based on thermodynamical conceptsthat can be found in Hillert [8]. They obtain a thermodynamically based procedure topredict non-equilibrium interface kinetics by using both analytical and numerical tech-niques.

However, all these mentioned models did not consider the technologically importantdissolution of multi-component particles in multi-component alloys. As far as we knowAgren et al. [9] was the first to extend the models to multi-component alloys. Hisformalism was based on a thermodynamic treatment of diffusion in terms of chemicalpotentials and an interface motion from a material balance. The numerical methods that

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were used by Agren were improved by Crusius et al. [10] and the diffusion model wasimproved in [11], which forms the backbone of the software-package DICTRA suitablefor dissolution and growth problems with one spatial dimension. The thermodynamicrelation, which defined the boundary conditions at the moving interface, was simplifiedto a hyperbolic relationship. This has been done for iron-based alloys by Vitek et al.

[12] and Hubert [13]. Furthermore, Reiso et al. [14] investigated the dissolution of Mg2Sialloys in aluminium alloys by the use of the same principles.

The above mentioned authors viewed particle dissolution as a Stefan problem: a dif-fusion equation with a sharp moving interface. A recent approach is the phase-fieldapproach, which is derived from a minimization of the energy functional and based ona diffuse interface between the consecutive phases. This approach has, among others,been used by Kobayashi et al. [15] and Burman et al. [16] to simulate dendritic growth.An extension to multi-component phase-field computation is done by Grafe et al. [17],where solidification and solid-state transformation is modelled. For the one-dimensionalcase they obtain a perfect agreement between the phase-field approach and the softwarepackage DICTRA, which is based on a sharp interface between the consecutive phases.Furthermore, some recent comparison studies of phase-field methods with a Stefan prob-lem, solved by a moving mesh method or a level-set method or a mesh-free method, weredone by Javierre et al. [18] and Kovacevic and Sarler [20] respectively. Some disadvan-tages of the phase-field approach are that (1) no simple quick estimation of the solutionis available, and that (2) physically justifiable parameters in the energy functional arenot easy to obtain. Generally those parameters are to be obtained from fitting proce-dures that link experiment, thermodynamic data-bases and numerical computation. Another disadvantage of the phase-field method is the requirement of a fine grid resolu-tion in the diffuse interface region in order to have agreement with the solution of the’sharp interface problem’. This poses a severe time-step criterion and hence time con-suming computations. This was observed by Burman et al. [16] and Javierre et al. [18].Therefore, we limit ourselves here to viewing particle dissolution as a (vector) Stefanproblem. We remark that Thornton et al. [21] wrote a nice review paper on simulatingdiffusional phase transformations using several physical model approaches as the updatedthermodynamic methods by Agren, used in the package DICTRA, and the diffuse inter-face phase-field and (Allen)-Cahn-(Hilliard) approach. Thornton et al. [21] also describeseveral two-dimensional applications of phase coarsening with Ostwald Ripening usingthe diffuse interface approach. The present paper will focus on the computational aspectsof solving Stefan problems with a sharp interface applied to particle dissolution in (multi-component) alloys. For a review on the solution of Stefan problems in multi-componentalloys, we refer to [28]. Furthermore, some mathematical issues will be summarized.

Although much work on the mathematical modeling of dissolution of particles had beendone, some major limitations remained (until recently):

1. No fast and efficient numerical method for the simultaneous dissolution of a particleand a segregation at the grain boundary was available. Furthermore, no quick

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and well motivated self-similar solution for the dissolution of particles in multi-component has been reported.

2. Some particles may be disk-shaped, hence a two-dimensional model is necessaryto compute the dissolution of the particle. With the classical literature of FiniteElements the computation of the interface movement with a sharp angle within theboundary is impossible. Furthermore, the case where two particles dissolve in onecell needs to be discussed.

3. No numerical model, be it 2D or quasi 3D, that treats the interface movement whilethe interfacial reactions take place exists.

4. Metallurgical experiments on alloys indicate that dissolving particles or phases maybreak up into smaller particles or phases in some circumstances. No metallurgicallysound model in three spatial dimensions, based on the sharp interface approach,exists to deal with these topological changes.

These limitations where lifted in a suite of mathematical models of increasing complexity.This paper presents a coherent total picture of the basic concepts and equations in thesemodels and illustrates their potential.

Furthermore, an experimental validation of the above mentioned models can be foundin [22] and [23]. In the first paper the activation energy for particle dissolution hasbeen analyzed. In the second paper the experimental validation was carried out usingDSC-measurements. New work concerns the analysis of particle dissolution where cross-diffusion aspects and a relaxation of thermodynamic equilibrium, are incorporated. Fur-ther, the level-set method and moving grid method are analyzed as candidates to modelparticle growth. In this paper we only consider the moving grid method in more detailin which the application concerns the growth of a particle under the influence of surfaceenergy effects.

1.3 Organization of the paper

The current paper does not aim at being mathematically rigorous but merely aimsat being descriptive about the implications of the developed mathematics of these morecomplex models. First we formulate the mathematical models for particle dissolution inSection 2. Here a one-dimensional multi-component model and two-dimensional model isformulated. Section 2 ends with a brief description of the mathematical implications ofthe models. Section 3 starts with available self-similar solutions for the one-dimensional(multi-component) models. Next, the numerical methods to solve the one- and two-dimensional models are described. Section 5 deals with applications of the models particlegrowth influenced by surface energy effects. Finally, some concluding remarks about thework are given and ongoing research is indicated.

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2 MODELS

We consider a particle of β phase surrounded by a diffusive phase α of either uniform ornonuniform composition. The boundary between the adjacent phases is referred to as theinterface. The metal is divided into representative cells in which a single particle of phaseβ dissolves in an α phase. Particle dissolution is assumed to proceed via the followingconsecutive steps: decomposition of the chemical bonds in the dissolving phase, crossingof the interface by the atoms and finally long-distance diffusion in the diffusive phase. Inthe present paper we assume that the rate of the dissolution process is determined bythe last step, e.g. at the interface between the adjacent phase we have local equilibrium.Hence the interface concentration are those as predicted by thermodynamics.

In this section, the model for a binary alloy and a multi-component alloy are presentedconsecutively.

2.1 The model for a binary alloy

We denote the interface, consisting of a point, curve or a surface for respectively a one-,two- or three-dimensional domain of computation by S = S(t). The fixed boundaries ofthe domain of computation are denoted by Γ. Further, the domain of computation is splitinto the diffusive part (the α-diffusive phase), denoted by Ω and the β-particle Ωp. Thedistribution of the alloying element is determined by diffusion in the diffusive phase Ω,which gives

∂c

∂t= D∆c = D div grad c, for x ∈ Ω(t) and t > 0. (1)

Here D represents the diffusion coefficient and x denotes the spatial position within thedomain of computation. In the present study we treat D as a constant. Within theparticle the concentration is equal to a given constant, hence

c = cpart, for x ∈ Ωp(t) and t ≥ 0. (2)

On the interface, S(t), we have local equilibrium, i.e. the concentration is as predicted bythe thermodynamic phase diagram, i.e.

c = csol, for x ∈ S(t) and t > 0. (3)

Further, at a boundary not being the moving interface, Γ, we have no flux of atoms, i.e.

D∂c

∂n= 0, for x ∈ Γ(t) and t > 0. (4)

In the above equation∂c

∂ndenotes the outward normal derivative of c on the fixed boundary

Γ. From a local mass balance, the equation of motion of the interface can be derived, thisequation is commonly referred to as the Stefan condition, and is given by:

(cpart − csol)vn = D∂c

∂n, for x ∈ S(t) and t > 0. (5)

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Here vn denotes the normal component of the interface velocity outward from Ω. Theproblem is completed with the initial concentration c0 and the initial position of theinterface S(0). The problem, consisting of equations (1), (2), (3), (4) and (5) is referredto as a Stefan problem for particle dissolution or particle growth. For the case of growthof particles, the Gibbs-Thomson effect is important in the early stages of growth afternucleation. The earliest-stage phenomenon of nucleation differs totally from the presentStefan problem. Models for nucleation are commonly based on Monte-Carlo simulationsor on evolution of the statistical particle size distribution (see for instance the work ofMyhr and Grong [30], where a hyperbolic partial differential equation is solved for theprobability density of the particle size). Therefore, we do not consider nucleation in thepresent study. Future studies will be devoted to the Gibbs-Thomson effect. For now, werefer to Mullins and Sekerka who show that spherical particles can only preserve theirspherical shape during growth if the Gibbs-Thomson effect is included (i.e. the Mullins-

Sekerka instability [31], [32]). We will only show some preliminary finite element resultsin the present paper, which confirm this behavior numerically.

2.2 The model for a multi component alloy

We consider a particle of a multi-component β phase surrounded by a ’matrix’ of phaseα, of either uniform or non-uniform composition. The boundary between the β-particleand α-matrix is referred to as the interface. The metal is divided into representativecells in which a single particle of phase β dissolves in an α-matrix. Particle dissolutionis assumed to proceed by a number of subsequent steps [4, 33]: decomposition of thechemical bonds in the particle, crossing of the interface by atoms from the particle andlong-distance diffusion in the α-phase. In the models of thermodynamic equilibrium, weassume in this section that the first two mechanisms proceed sufficiently fast with respectto long-distance diffusion and do not affect the dissolution kinetics. Hence, the interfacialconcentrations are those as predicted by thermodynamics (local equilibrium). Later thisassumption is dropped. In [34] we considered the dissolution of a stoichiometric particlein a ternary alloy. The hyperbolic relationship between the interfacial concentrationsfor ternary alloys is derived using a three-dimensional Gibbs space. For the case thatthe particle consists of n chemical elements apart from the atoms that form the bulkof the β-phase, a generalization to a n-dimensional Gibbs hyperspace has to be made.The Gibbs surfaces become hypersurfaces. We expect that similar consequences applyand that hence the hyperbolic relation between the interfacial concentrations remainsvalid for the general stoichiometric particle in a multi-component alloy. We denote thechemical species by Spi, i ∈ 1, ..., n+1. We denote the stoichiometry of the particle by(Sp1)m1

(Sp2)m2(Sp3)m3

(...)(Spn)mn. The numbers m1, m2, ... are stoichiometric constants.

We denote the interfacial concentration of species i by csoli and we use the following

hyperbolic relationship for the interfacial concentrations:

f(csol1 , csol

2 , . . . , csoln ) = (csol

1 )m1(csol2 )m2(...)(csol

n )mn − Ksol = 0, Ksol = Ksol(T ). (6)

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The factor Ksol is referred to as the solubility product. It depends on temperature Taccording to an Arrhenius relationship. In principle, the model can handle any form oftemperature dependence for the solubility product.

We denote the position of the moving interface between the β-particle and α-phase byS(t). Consider a one-dimensional domain, i.e. there is one spatial variable, which extendsfrom 0 up to M . Since particles dissolve simultaneously in the metal, the concentrationprofiles between consecutive particles may interact and hence soft-impingement occurs.This motivates the introduction of finitely sized cells over whose boundary there is no flux.For cases of low overall concentrations in the alloy, the cell size M may be large and thesolution resembles the case where M is infinite. The latter case can be treated easily with(semi) explicit expressions. The spatial co-ordinate is denoted by r, 0 ≤ S(t) ≤ r ≤ M .The α-matrix where diffusion takes place is given by Ω(t) := r ∈ R : 0 ≤ S(t) ≤ r ≤ M.The β-particle is represented by the domain 0 ≤ r < S(t). Hence for each alloying element,we have for r ∈ Ω(t) and t > 0 (where t denotes time)

∂ci

∂t=

n∑

j=1

Dij∆cj, for i ∈ 1, ..., n. (7)

Here Dij and ci respectively denote the (cross-)diffusion coefficients and the concentrationof the species i in the α-rich phase. If Dij < 0 for some i 6= j, then, the transport of elementi is delayed by the presence of element j. For Dij > 0, the opposite holds. Experimentswith Differential Scanning Calorimetry by Chen et al. [23] for Al-Si-Mg alloys indicate thatdisregarding cross-diffusion terms gives a good approximation. However, for some otheralloys the full diffusion matrix should be taken into account. A physical motivation of theabove partial differential equation is given by Kirkaldy and Young [35]. The geometry isplanar, cylindrical and spherical for respectively a = 0, 1 and 2. Let c0

i denote the initialconcentration of each element in the α phase, i.e. we take as initial conditions (IC) forr ∈ Ω(0)

(IC)

ci(., 0) = c0i (.) for i ∈ 1, ..., n

S(0) = S0.

At a boundary not being an interface, i.e. at Γ or when S(t) = 0, we assume no fluxthrough it, i.e.

∂ci

∂n= 0, for i ∈ 1, ..., n. (8)

Furthermore at the moving interface S(t) we have the ’Dirichlet boundary condition’ csoli

for each alloying element. The concentration of element i in the particle is denoted bycparti , this concentration is fixed at all stages. This assumption follows from the constraint

that the stoichiometry of the particle is maintained during dissolution in line with Reisoet al. [14]. The dissolution rate (interfacial velocity) is obtained from a mass-balance.

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Summarized, we obtain at the interface for t > 0 and i, j ∈ 1, ..., n:

ci = csoli for x ∈ S(t),

vn =

n∑

j=1

Dij

cparti − csol

i

∂cj

∂nfor x ∈ S(t)

⇒n

∑

k=1

Dik

cparti − csol

i

∂ck

∂n=

n∑

k=1

Djk

cpartj − csol

j

∂ck

∂nfor x ∈ S(t).

(9)The right part of above equations follows from local mass-conservation of the components.Above formulated problem falls within the class of Stefan-problems, i.e. diffusion with amoving boundary. Since we consider simultaneous diffusion of several chemical elements,it is referred to as a ’vector-valued Stefan problem’. The unknowns in above equationsare the concentrations ci, interfacial concentrations csol

i and the interfacial position S(t).All concentrations are non-negative. The model was analyzed in [26, 27, 25].

The influence of cross-diffusion is investigated in terms of a parameter study in [25] andin terms of self-similar solutions as exact solutions for the unbounded domain in [26]. Ananalysis of several numerical analysis in terms of stability criteria is presented in [27].For a mathematical overview of Stefan problems we refer to the textbooks of Crank [6],Chadam and Rasmussen [36] and Visintin [37].

3 ANALYTIC SOLUTIONS

The early models on particle dissolution and growth based on long-distance diffusionconsisted of analytic solutions in an unbounded medium under the assumption of localequilibrium at the interface, see Ham [38, 39], Zener [40], Whelan [1] and Aaron and Kotler[5] to mention a few. These analytic models have been developed for idealized geometriessuch as planar, cylindrical and spherical geometries and are mostly based on the so-calledBoltzmann transformation where the concentration c is written as c(x, t) = c(η) whereη = x/

√t. Recently the self-similar solutions have been extended to multi-component

alloys by Atkinson et al. [41] and Vermolen et al. [26]. In the latter paper an effectivediffusion coefficient was determined which is useful for a quick estimate of the dissolutionor growth speed. Recently, the Zener solutions for the various geometries have been ex-tended to multi-component alloys [42]. We remark that these solutions hold for the caseof local equilibrium at the moving interface, that is, the concentrations at the interfaceare those as given by the phase diagrams.

The model of Nolfi et al. [4] incorporates the interfacial reaction between the dissolvingparticle and its surrounding phase by the use of a Fourier series solution for the concen-tration profile. The interface motion was not included in this solution. Vermolen et al.

[43] constructed a semi-analytic solution based on Fourier Series in which the interfacemotion was incorporated. Further solutions with Laplace transforms have been reportedby Aaron & Kotler [5] and Whelan [1]. These solutions were extended to multi-component

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alloys by Reiso et al. [14]. The analytic solutions are very valuable for the evaluation ofthe accuracy of the numerical procedures.

4 NUMERICAL SOLUTION PROCEDURES

In the literature one can find various numerical methods to solve Stefan problems.These methods can be distinguished in the following categories: front-tracking, front-fixing and fixed-domain methods. In a front-fixing method a transformation to body fittedcurvilinear coordinates is used (a special case is the Isotherm Migration Method (IMM)[6]). A drawback is that such a transformation can only be used for a relatively simplegeometry. Fixed-domain methods are the enthalpy method (EM), and the variationalinequality method (VI). In these methods a new unknown is introduced, which is theintegral of the primitive variable. The free boundary is implicitly defined by this unknown.Since in our approach the equations hold for the concentration and there are no energiesinvolved in the model, the enthalpy method and phase-field method are not used. Werefer to [44] and [45] among others where the phase-field method is used to computethe solution of the moving boundary problem. An other recent method where the freeboundary is implicitly defined is the level-set method as described by Chen et al. [46]for Stefan problems. Here the interface is identified by the zero level-set of a markerfunction. The advantages of this method is that topological changes, such as breaking up,of the dissolving or growing phases are dealt with easily. On the other hand, since both theinterfacial velocity and interfacial concentrations are here determined by the concentrationgradient, a grid grid-refinement near the free boundary can be attractive. This impliesthat the grid moves anyway and hence the benefits for the level-set method due to afixed grid no longer apply. Though, the level-set method remains the best candidatedue to the ability of dealing with changing topologies and because remeshing steps arenot needed. The IMM and VI methods are only applicable when the interface is an equi-concentration line. However, in our application where either multi-component particles orinterfacial reactions are taken into account, the interface is not an equi-concentration line.Hence, (IMM) and (VI) methods are no suitable candidates. Therefore we use a front-tracking method which has the added benefit that a first order reaction at the interfacecan be incorporated in the model. The moving grid method solves one partial differentialequation only. The mesh is moved using an arbitrary Lagrangian Eulerian method. Here,the method is relatively cheap compared to the level-set method, where also a first orderhyperbolic partial differential equation for the level set function has to be solved with acontinuous extension of the interface velocity at each time-step. However, it is sometimesnecessary to change the topology of the elements, then, the moving grid method requiresa remeshing step, which involves an expensive two dimensional interpolation step. Thisis a very expensive step in the moving grid method. Further, topological changes ofgrowing and dissolving (for instance the dissappearrance) phases are hard to implementinto moving mesh methods. For these cases the level-set method becomes more attractive.We refer to [27, 47, 19] for more details on our level set method and moving grid method.

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Further, for review papers we refer to Thornton et al. [21] and Vermolen et al. [28]. Thefirst paper focusses more on an incorporation of the relevant thermodynamics and diffuseinterface models. The second paper deals with numerical aspects of the solution of thesharp interface model. A recent book on the level set method and its applications is dueto Osher and Fedkiw [29].

5 APPLICATION OF MOVING GRID TO CURVATURE DEPENDENCE

In the present application we consider a binary alloy in which a particle dissolves orgrows. The calculations that we present here are for two spatial dimensions. Due to crystaldefects, such as dislocations, the free energy of the phases present is raised. This causesthe so-called Gibbs-Thomson effect, which makes the interface concentration sensible tothe local curvature. This effect was not taken into account in the previous section. Letcsol0 be the equilibrium interface concentration that follows from the phase diagram, then,

the interface concentration at a location at the interface with curvature κ, is given by

csol = csol0 exp

(

2γVmκ

RT

)

, for x ∈ S(t). (10)

This is an important effect if the curvature of the particle is small. Here γ, Vm and Rrespectively denote the surface energy, molar volume and ideal gas constant. The aboveequation requires the determination of the curvature κ on each location at the movinginterface S(t). The present approach solves second Fick’s Law for the concentration witha moving grid method to model the movement of the boundary. Using a finite elementmethod to solve the equations with a moving grid, that is, for the concentration we solvethe following problem:

Find c(x, t) ∈ H1(Ω(t)), subject to c(x, t) = csol(x, t) for x ∈ S(t),

such that

∫

Ω(t)

∂c

∂t− vmesh · ∇c

v(x)dΩ = −∫

Ω(t)

∇c · ∇vdΩ,

for all v ∈ H1(Ω(t)), subject to v = 0 for x ∈ S(t).

(11)

In the above equation vmesh = x(t+∆t)−x(t)∆t

is the mesh velocity. Linear triangular elementsare used for the discretization. The displacement of the moving boundary is computedin a mass conserving manner, see [47]. To determine the interface curvature, we usethe observation that three points uniquely define the circle containing these points, unlessthese points are on a straight line. The radius of this circle is the local radius of curvature,of which its reciprocal in turn defines its local curvature. This local curvature is used forthe determination of the local interface concentration, see equation (10). The curvatureis allowed to be negative whenever the interface is concave-upward with respect to theparticle. For other cases the curvature is positive. A sketch is presented in Figure 1. A

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consequence of the Gibbs-Thomson effect is that the fingers with smaller (or negative)curvature have a larger growth rate than fingers with a higher curvature. Hence, theGibbs-Thomson effect prevents positive fingers from moving rapidly into the matrix andto accelerate the movement into the matrix of the negative fingers.

desired

speed

curvature

negativecurvature positive

Figure 1: Positive and negative curvature and desired speed.

The second derivative is needed for the local curvature. Further, the sign of the vectorproduct of the line segments connecting the adjacent points is used to determine the signof the curvature. This is all subsitituted into equation (10). The numerical determinationof the second derivative is a notorious problem in computing the local curvature due torounding errors.

12V

23V

12V23V

1P

2P

positive

3P

3P

2P

1P

12V 23V^ < 0 12V 23V^ > 0

negative curvature curvature

Figure 2: Positive and negative curvature.

A problem that occured here was the fact that a small error in the curvature, gives arelatively large influence on the interface concentration due to its exponential dependence(see equation (10)). This gives unrealistic results. Since this is a numerical artefact, weremoved it by introducing a ’cut-off’ procedure, by adjusting the interface concentrationto the average of the interface concentration and the initial concentration. The choice issomewhat arbitrary, but the results become more realistic. The ’cut-off’ procedure givesthat

If csol > c0 then csol → csol + c0

2.

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A next idea was to smooth the curvature by using an approximating spline. This im-plies that for the determination of the local curvature, the interface points are somewhatshifted so that the interface becomes smoother. The second derivative was approximatedsubsequently. Our preliminary numerical experiments look promising. Additional re-search on this idea is needed.

Finally, we apply the idea of determining the local radius of curvature by using a circleintersecting three adjacent points as discussed in combination with the ’cut-off’ procedure.The cut-off procedure is combined with the decrease of the time-step when the mesh isrefined. Here we got convergence for decreasing time-step and mesh-width.

As an application we consider the growth of a circular particle with D = 0.1 µm/s,c0 = 0.5, cpart = 1, csol

0 = 0.0001 and initial radius of 0.3 µm. These data imply thatthe particle will grow. As Mullins & Sekerka [31] point out, the growth will occur underfingering if the surface energy γ is lower than a threshold value (see Figure 3). To illustratethe influence of the surface energy γ we plot the results for the case that 2γVm

RT= 2.406,

which is a value for which no fingering occurs during growth, in Figure 4. It can be seenthat no fingering occurs, which is in line with the expectations from the theory of Mullins& Sekerka.

Figure 3: Growth modeled with γ = 0. Figure 4: Growth modeled with 2γVm

RT= 2.406.

From this we see that the statement due to Mullins & Sekerka that the Gibbs-Thomsoneffect prevents the occurrence of fingering of a growing particle is also supported numeri-cally. For more details concerning this study on the issue of fingering, we refer to Pinson[48].

6 CONCLUSIONS

A suite of mathematical models for the diffusional phase transformations has beenpresented. Some applications with a two-dimensional model incorporating the Gibbs-Thomson effect are shown for a binary alloy. The Gibbs-Thomson effect can be used toprevent fingering of a growing particle.

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A SUITE OF SHARP INTERFACE MODELS FOR SOLID …ta.twi.tudelft.nl/nw/users/vuik/papers/Ver06PVZ.pdf · A SUITE OF SHARP INTERFACE MODELS FOR ... Abstract. A suite of mathematical models